Domain decomposition preconditioning for discontinuous galerkin approximations of convection-diffusion problems

Abstract

In the classical Schwarz framework for conforming approximations of nonsymmetric and indefinite problems [5, 6] the finite element space is optimally decomposed into the sum of a finite number of uniformly overlapped, two-level subspaces. In each iteration step, a coarse mesh problem and a number of smaller linear systems, which correspond to the restriction of the original problem to subregions, are solved instead of the large original system of equations. Based on this decomposition, domain decomposition methods of three basic type-additive, multiplicative and hybrid Schwarz methods-have been studied in the literature (cf. [4, 5, 6]). In [1, 2] it was shown that for discontinuous Galerkin (DG) approximations of purely elliptic problems optimal nonoverlapping Schwarz methods (which have no analogue in the conforming case) can be constructed. Moreover, it was proved that they exhibit spectral bounds analogous to the one obtained with conforming finite element approximations in the case of "small" overlap, making Schwarz methods particularly well-suited for DG preconditioning. Motivated by the above considerations, we study a class of nonoverlapping Schwarz preconditioners for DG approximations of convection-diffusion equations. The generalized minimal residual (GMRES) Krylov space-based iterative solver is accelerated with the proposed preconditioners.We discuss the issue of convergence of the resulting preconditioned iterative method, and demonstrate through numerical computations that the classical Schwarz convergence theory cannot be applied to explain theoretically the converge observed numerically. © 2009 Springer-Verlag Berlin Heidelberg.